Full characterizations of the variational McShane Integral on $m$-dimensional compact intervals

TL;DR

This paper provides a complete characterization of the variational McShane integral for functions defined on multi-dimensional intervals, using additive interval functions and convex average ranges.

Contribution

It introduces necessary and sufficient conditions for additive interval functions to be primitives of variational McShane integrable functions in multiple dimensions.

Findings

Characterization of variational McShane integrability in higher dimensions.

Conditions based on convex cubic average range of additive functions.

Extension of integral theory to multi-dimensional compact intervals.

Abstract

In this paper we consider the additive interval functions defined on the family $\mathcal{I}_{m}$ of all non-degenerate closed subintervals of the cubic interval $C_{m} = [0,1]^{m}$ in the $m$-dimensional Euclidean space $\mathbb{R}^{m}$ and taking values in a Banach space $X$. We give necessary and sufficient conditions for an additive interval function $F : \mathcal{I}_{m} \to X$ to be the primitive of a variational McShane (or strong McShane) integrable function $f : C_{m} \to X$ in terms of the convex cubic average range of $F$.